Reciprocity relationships for gravitational-wave interferometers Yuri Levin (Monash University) 1. Example: creep noise 2. Formalism 3. Creep noise again 4. Thermal deformations of mirrors 5. Thermal noise 6. Opto-mechanical displacements 7. Discussion

Part 1: creep noise Ageev et al. 97 Cagnoli et al.97 De Salvo et al. 97, 98, 05, 08

Quakes in suspension fibers defects Sudden localized stress release: non-Gaussian (probably), statistics not well-understood, intensity and frequency not well-measured. No guarantee that it is unimportant in LIGO II or III Standard lore: couples through random fiber extension and Earth curvature. KAGRA very different b/c of inclined floor Much larger direct coupling exists for LIGO. Top and bottom defects much more important. Levin 2012

Part 2: Reciprocity relations If you flick the cow’s nose it will wag its tail. If someone then wags the cow’s tail it will ram you with its nose. Provided that the cow is non-dissipative and follows laws of elastodynamics the coupling in both directions is the same

Reciprocity relations Force density Readout variable displacement form-factor

Reciprocity relations Force density Readout variable displacement form-factor is invariant with respect to interchange of and

stress Part 3: the creep noise again

The response to a single event: Location of the creep event Pendulum mode Violin mode

Random superposition of creep events parameters, e.g. location, volume, strength of the defect. Fourier transform Probability distribution function Caveat: in many “crackle noise” system the events are not independent

Conclusions for creep: Simple method to calculate elasto-dynamic response to creep events Direct coupling to transverse motion Response the strongest for creep events near fibers’ ends => Bonding!

Part 4: thermal deformations of mirrors High-temperature region Not an issue for advanced KAGRA. Major issue for LIGO & Virgo Zernike polynomials New coordinates cf. Hello & Vinet 1990 Treat this as a readout variable

How to calculate Apply pressure to the mirror face Calculate trace of the induced deformation tensor Have to do it only once! Calculate the thermal deformation Young modulus Thermal expansion Temperature perturbation King, Levin, Ottaway, Veitch in prep.

Check: axisymmetric case (prelim) Eleanor King, U. of Adelaide

Off-axis case (prelim) Eleanor King, U. of Adelaide

Part 5: thermal noise from local dissipation Readout variable Conjugate pressure Uniform temperature Local dissipation Non-uniform temperature. Cf. KAGRA suspension fibers See talk by Kazunori Shibata this afternoon

Part 6: opto-mechanics with interfaces Question: how does the mode frequency change when dielectric interface moves? Theorem: Mode energy Interface displacement Optical pressure on the interface Useful for thermal noise calculations from e.g. gratings (cf. Heinert et al. 2013)

Part 6: opto-mechanics with interfaces Linear optical readout, e.g. phase measurements Carrier light + Perturbation PhaseForm-factor

Part 6: opto-mechanics with interfaces Linear optical readout, e.g. phase measurements Photo-diode PhaseForm-factor

Part 6: opto-mechanics with interfaces Photo-diode 1. Generate imaginary beam with oscillating dipoles 2. Calculate induced optical pressure on the interface 3. The phase

Conclusions Linear systems (elastic, optomechanical) feature reciprocity relations They give insight and ensure generality They simplify calculations